**Types of set : **

The concept of set is vital to mathematical thought and is being used in almost every branch of mathematics. In mathematics, sets are convenient because all mathematical structures can be regarded as sets.

Here, we are going to see the different types of set.

A set containing no elements is called the empty set or null set or void set.

**Reading notation :**

So, it is denoted by { } or ∅

For example,

Consider the set A = { x : x < 1, x ∈ N }

There is no natural number which is less than 1.

Therefore, A = { }, n(A) = 0.

Note : The concept of empty set plays a key role in the study of sets just like the role of the number zero in the study of number system.

Let us look at the next stuff on "Types of set"

If the number of elements in a set is zero or finite, then the set is called a finite set.

For example,

(i) Consider the set A of natural numbers between 8 and 9.

There is no natural number between 8 and 9.

So, A = { } and n(A) = 0.

Hence, A is a finite set.

(ii) Consider the set X = { x : x is an integer and -1 ≤ x ≤ 2 }

So, X = { -1, 0, 1, 2 } and n(X) = 4

Hence, X is a finite set.

Note : The cardinal number of a finite set is finite.

Let us look at the next stuff on "Types of set"

A set is said to be an infinite set if the number of elements in the set is not finite.

For example,

Let W = The set of all whole numbers .

That is, W = { 0, 1, 2, 3, ......................}

The set of all whole numbers contain infinite number of elements.

Hence, W is an infinite set.

Note : The cardinal number of an infinite set is not a finite number.

Let us look at the next stuff on "Types of set"

A set containing only one element is called a singleton set.

For example,

Consider the set A = { x : x is an integer and 1 < x < 3 }

So, A = { 2 }. That is, A has only one element.

Hence, A is a singleton set.

Note : { 0 } is not null set. Because it contains one element. That is "0".

Let us look at the next stuff on "Types of set"

Two sets A and B are said to be equivalent if they have the same number of elements.

In other words, A and B are equivalent if n(A) = n(B).

**Reading notation :**

"A and B are equivalent" is written as A ≈ B

For example,

Consider A = { 1, 3, 5, 7, 9 } and B = { a, e, i, o, u }

Here n(A) = n(B) = 5

Hence, A and B are equivalent sets.

Let us look at the next stuff on "Types of set"

Two sets A and B are said to be equal if they contain exactly the same elements, regardless of order.

Otherwise the sets are said to be unequal.

In other words, two sets A and B are said to be equal if

(i) every element of A is also an element of B and

(ii) every element of B is also an element of A.

**Reading notation :**

For example,

Consider A = { a, b, c, d } and B = { d, b, a, c }

Set A and set B contain exactly the same elements.

And also n(A) = n(B) = 4.

Hence, A and B are equal sets.

Note : If n(A) = n(B), then the two sets A and B need not be equal. Thus, equal sets are equivalent but equivalent sets need not be equal.

Let us look at the next stuff on "Types of sets"

A set X is a subset of set Y if every element of X is also an element of Y.

In symbol we write

**x ⊆ y**

**Reading Notation :**

Read ⊆ as "X is a subset of Y" or "X is contained in Y"

Read ⊈ as "X is a not subset of Y" or "X is not contained in Y"

Let us look at the next stuff on "Types of set"

A set X is said to be a proper subset of set Y if X ⊆ Y and X ≠ Y.

In symbol, we write X ⊂ Y

**Reading notation :**

Read X ⊂ Y as "X is proper subset of Y"

The figure given below illustrates this.

The set of all subsets of A is said to be the power set of the set A.

**Reading notation :**

The power set of A is denoted by P(A)

A set X is said to be a proper subset of set Y if X ⊆ Y and X ≠ Y.

In symbol, we write X ⊂ Y

Here,

**Y is called super set of X **

If A is the given set and it contains "n" number of elements, we can use the following formula to find the number of subsets.

**Number of subsets = 2ⁿ**

**Formula to find the number of proper subsets :**

**Number of proper subsets = ****2ⁿ****⁻¹**

We already know that the set of all subsets of A is said to be the power set of the set A and it is denoted by P(A).

If A contains "n" number of elements, then the formula for cardinality of power set of A is

**n[P(A)] = 2ⁿ**

**Note :**

Cardinality of power set of A and the number of subsets of A are same.

Null set is a proper subset for any set which contains at least one element.

For example, let us consider the set A = { 1 }

It has two subsets. They are { } and { 1 }.

Here null set is proper subset of A. Because null set is not equal to A.

If null set is a super set, then it has only one subset. That is { }.

More clearly, null set is the only subset to itself. But it is not a proper subset.

Because, { } = { }

Therefore, A set which contains only one subset is called null set.

To have better understand on "Subset of a given set", let us look some examples.

**Example 1 :**

Let A = {1, 2, 3, 4, 5} and B = { 5, 3, 4, 2, 1}. Determine whether B is a proper subset of A.

**Solution : **

If B is the proper subset of A, every element of B must also be an element of A and also B must not be equal to A.

In the given sets A and B, every element of B is also an element of A. But B is equal A.

**Hence, B is the subset of A, but not a proper subset. **

**Example 2 :**

Let A = {1, 2, 3, 4, 5} and B = {1, 2, 5}. Determine whether B is a proper subset of A.

**Solution : **

If B is the proper subset of A, every element of B must also be an element of A and also B must not be equal to A.

In the given sets A and B, every element of B is also an element of A.

And also But B is not equal to A.

**Hence, B is a proper subset of A. **

**Example 3 :**

Let A = {1, 2, 3, 4, 5} find the number of proper subsets of A.

**Solution : **

Let the given set contains "n" number of elements.

Then, the formula to find number of proper subsets is

**= ****2ⁿ****⁻¹**

The value of "n" for the given set A is "5".

Because the set A = {1, 2, 3, 4, 5} contains "5" elements.

Number of proper subsets = 2⁵⁻¹

= 2⁴

= 16

**Hence, the number of proper subsets of A is 16.**

**Example 4 :**

Let A = {1, 2, 3 } find the power set of A.

**Solution : **

We know that the power set is the set of all subsets.

Here, the given set A contains 3 elements.

Then, the number of subsets = 2³ = 8

Therefore,

**P(A)** = **{**** {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}, { }**** }**

**Example 5 :**

Let A = {a, b, c, d, e} find the cardinality of power set of A

**Solution : **

The formula for cardinality of power set of A is given below.

**n[P(A)] = 2ⁿ**

Here "n" stands for the number of elements contained by the given set A.

The given set A contains "5" elements. So n = 5.

Then, we have

n[P(A)] = 2⁵

n[P(A)] = 32

**Hence, the cardinality of the power set of A is 32. **

After having gone through the stuff given above, we hope that the students would have understood "Types of set".

Apart from the stuff given above, if you want to know more about "Types of set", please click here

Apart from the stuff, "Types of set", if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**

**Sum of all three four digit numbers formed using 0, 1, 2, 3**

**Sum of all three four digit numbers formed using 1, 2, 5, 6**